Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62993 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2);(4,8)(9,10)(11,12,16,17)(13,18,23,26)(14,19,24,22)(15,20,25,21);(3,4)(5,9)(6,10)(7,8);(12,17)(15,25)(18,26)(19,22);(3,5,7,6)(4,9,8,10)(11,13)(12,18)(14,20)(15,22)(16,23)(17,26)(19,25)(21,24);(3,6,7,5)(4,10,8,9)(11,14)(12,19)(13,21)(15,18)(16,24)(17,22)(20,23)(25,26);(11,15)(12,20)(13,22)(14,18)(16,25)(17,21)(19,23)(24,26)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493233);